Statistics and probability

Ten customers at a bank recorded how long, in minutes, they waited to be served.
12, 7, 15, 9, 12, 18, 10, 12, 6, 14

A football team played 24 matches. The number of goals the team scored in each match is shown below.
Goals scored: 0, 1, 2, 3, 4
Frequency: 5, 8, 6, 3, 2

At Hillcrest School, 180 students were asked to name their favourite sport.
Football: 70 students, Tennis: 35 students, Swimming: 45 students, Other: 30 students.
The results are to be shown in a pie chart.

The stem-and-leaf diagram gives the scores of 15 students in a quiz.
2 | 5 8 9
3 | 1 3 3 7
4 | 0 2 4 4 4 8
5 | 3 6
Key: 3 | 1 means 31 points

A basketball player scored a mean of 8 points per game in her first 6 games of the season.
After her 7th game, her mean was 9 points per game.

A four-coloured spinner is biased. The table gives some of the probabilities of the spinner landing on each colour.

Red: 0.15   Blue: 0.4   Green: x   Yellow: x

The probability of landing on green is the same as the probability of landing on yellow.

A bag contains 7 red counters, 5 blue counters and 8 white counters only. A counter is taken at random from the bag.

Sam throws a drawing pin 250 times. It lands point up 95 times.

The probability that Lena's morning train is late on any day is 0.15. Whether the train is late on one day is independent of any other day. Lena catches the train on Monday and on Tuesday.

A bag contains 5 red counters and 3 blue counters only. Priya takes a counter at random from the bag, records its colour and puts it back in the bag. She then takes a second counter at random from the bag and records its colour.

Leila has a fair five-sided spinner numbered 1, 2, 3, 4 and 5, and a fair coin. She spins the spinner once and flips the coin once.

40 people each completed a puzzle. The time, t minutes, that each person took is summarised below.
0 < t ≤ 10: 4
10 < t ≤ 20: 9
20 < t ≤ 30: 15
30 < t ≤ 40: 8
40 < t ≤ 50: 4

Ravi recorded the waiting times, in minutes, of 7 customers at Coffee Shop A.
3.1, 4.5, 2.8, 5.0, 3.9, 4.2, 3.6
For Coffee Shop B, the median waiting time was 3.4 minutes and the range was 4.1 minutes.

Some students were asked how many pets they own.
Number of pets: 0, 1, 2, 3
Frequency: 7, x, 9, 4
The mean number of pets is 1.4.

Class A has 18 students. Their mean test mark is 64.
Class B has 12 students. Their mean test mark is 71.

Two fair six-sided dice, each numbered 1 to 6, are rolled.

There are 120 students in a year group. 52 are boys and the rest are girls. 18 of the boys wear glasses. In total, 85 students do not wear glasses.

Five positive whole numbers have
a mode of 4
a median of 6
a mean of 7
a range of 10.

A jar contains 11 sweets. 7 of the sweets are green and 4 are yellow. Kofi takes a sweet at random and eats it. He then takes a second sweet at random and eats it.

120 students each study exactly one of French or Spanish.
In Year 10, 28 students study French and 22 study Spanish.
In Year 11, 39 students study French and 31 study Spanish.
One of the 120 students is chosen at random.

The probability that it rains on a school day is 0.3.
If it rains, the probability that Ama walks to school is 0.2.
If it does not rain, the probability that Ama walks to school is 0.65.

The masses, m kg, of 80 parcels are recorded.

0 < m ≤ 2 : frequency 12
2 < m ≤ 5 : frequency 27
5 < m ≤ 10 : frequency 25
10 < m ≤ 20 : frequency 16

The information is to be shown in a histogram.

A histogram shows the heights, h cm, of some seedlings. The frequency densities of the bars are:

0 < h ≤ 4 : 2.5
4 < h ≤ 6 : 8
6 < h ≤ 10 : 4.5
10 < h ≤ 15 : 1.2

The lengths, L minutes, of 200 phone calls to a helpline are summarised.

L ≤ 2 : 28
L ≤ 4 : 76
L ≤ 6 : 132
L ≤ 8 : 170
L ≤ 10 : 188
L ≤ 12 : 200

The table gives information about the times, t minutes, that 50 customers spent waiting at a bank.

0 < t ≤ 10 : frequency 4
10 < t ≤ 20 : frequency 11
20 < t ≤ 30 : frequency 18
30 < t ≤ 40 : frequency 12
40 < t ≤ 50 : frequency 5

The probability that Amira's train is late on any day is 0.12. Whether the train is late on one day is independent of any other day.

The heights, h cm, of 80 plants are summarised below.
0 < h ≤ 10: 6
10 < h ≤ 20: 14
20 < h ≤ 30: 25
30 < h ≤ 40: 22
40 < h ≤ 50: 13

A histogram shows the commuting times, t minutes, of a group of workers. The frequency densities of the bars are:
0 < t ≤ 5: 2.4
5 < t ≤ 10: 4.0
10 < t ≤ 20: 3.2
20 < t ≤ 40: 0.8

A box contains 6 red pens and 4 green pens only. Mo takes two pens at random from the box without replacement.

A biased four-sided dice is numbered 1, 2, 3 and 4.
P(1) = 0.2, P(2) = x, P(3) = 2x, P(4) = 0.35

Each time Jade takes a basketball free throw, the probability that she scores is 0.7, independently of any other throw. Jade takes 3 free throws.

There are 30 students in a club. 17 play football, 12 play tennis and 6 play both football and tennis.

A bag contains x counters. 3 of the counters are red and the rest are green. Two counters are taken at random from the bag without replacement. The probability that both counters are red is 1/15.

A sports club has 50 members.
32 members play tennis.
25 members play squash.
6 members play neither tennis nor squash.

The probability that Sam hits a target with each arrow is 0.7, independently of other arrows. Sam keeps shooting until he hits the target, but he is allowed a maximum of three arrows.

The probability that Dev passes his driving test at the first attempt is 0.55.
If he fails, the probability that he passes at the second attempt is 0.7.
If he fails the second attempt, the probability that he passes at the third attempt is 0.8.

A sports club has 150 members. Their ages, a years, are grouped.

10 < a ≤ 20 : frequency not given
20 < a ≤ 30 : frequency 36
30 < a ≤ 50 : frequency not given
50 < a ≤ 80 : frequency 27

In a histogram of the data, the frequency density of the class 10 < a ≤ 20 is 2.4.

A histogram shows the masses, m grams, of apples from an orchard. The frequency densities are:

80 < m ≤ 100 : 0.6
100 < m ≤ 110 : 2.4
110 < m ≤ 120 : 3.0
120 < m ≤ 150 : 0.5

The lifetimes, h hours, of 200 batteries of brand X were tested.

h ≤ 30 : 10
h ≤ 35 : 50
h ≤ 40 : 110
h ≤ 45 : 170
h ≤ 50 : 190
h ≤ 60 : 200

Assume the cumulative frequency graph is made of straight lines joining the plotted points.
For brand Y, the median lifetime is 42 hours and the interquartile range is 15 hours.

A histogram shows the times, t minutes, that students spent on homework one evening. No scale is given on the frequency density axis.

The bar for 0 < t ≤ 5 is 3.6 cm tall and represents 27 students.
The bar for 5 < t ≤ 15 is 2.4 cm tall.
The bar for 15 < t ≤ 30 is 0.8 cm tall.

A box contains 10 balls. 6 of the balls are white and 4 are black. Three balls are taken at random from the box, one after another, without replacement.

The speeds, v mph, of cars passing a school were recorded. A histogram of the data has these frequency densities:

20 < v ≤ 30 : 1.2
30 < v ≤ 35 : 6.4
35 < v ≤ 40 : 8.8
40 < v ≤ 50 : 3.2
50 < v ≤ 70 : 0.4

A factory has two machines, A and B, that make light bulbs.
Machine A makes 60% of the bulbs and machine B makes the rest.
3% of the bulbs made by machine A are faulty.
5% of the bulbs made by machine B are faulty.
A bulb is chosen at random.

160 students sat a test marked out of 100. The cumulative frequency table shows their marks, x.

x ≤ 20 : 8
x ≤ 40 : 30
x ≤ 50 : 62
x ≤ 60 : 110
x ≤ 70 : 140
x ≤ 80 : 152
x ≤ 100 : 160

Assume the cumulative frequency graph is made of straight lines joining the plotted points.

A group of students contains g girls and b boys.
The mean mass of the girls is 52 kg.
The mean mass of the boys is 60 kg.
The mean mass of the whole group is 55.2 kg.

A bag contains n counters. 3 of the counters are red and the rest are blue. Two counters are taken at random from the bag without replacement. The probability that both counters are red is \(\frac{1}{15}\).

The probability that it rains on a given morning is 0.35.
If it rains, the probability that Tom is late for work is 0.4.
If it does not rain, the probability that Tom is late for work is 0.1.

Bag A contains 3 red counters and 5 blue counters.
Bag B contains 4 red counters and 2 blue counters.
A counter is taken at random from bag A and placed in bag B. A counter is then taken at random from bag B.

A box contains 12 pens: 5 black, 4 blue and 3 red. Ben takes two pens at random from the box without replacement.

A histogram shows the lengths, l cm, of fish caught in a lake. The frequency densities are:

20 < l ≤ 30 : 1.5
30 < l ≤ 35 : 5.2
35 < l ≤ 40 : 6.0
40 < l ≤ 50 : 1.6
50 < l ≤ 60 : 0.3

80 people were asked how long, t minutes, they spent travelling to work. The data are shown in a histogram.

• The class 0 < t ≤ 10 has frequency density 1.2
• The class 10 < t ≤ 15 has frequency density three times that of the class 0 < t ≤ 10
• The class 15 < t ≤ 25 has frequency x
• The class 25 < t ≤ 40 has frequency density 0.4

A histogram shows the waiting times, w minutes, of patients at a clinic. The frequency densities are:

0 < w ≤ 4 : 2.5
4 < w ≤ 6 : 9
6 < w ≤ 10 : 6.5
10 < w ≤ 20 : 0.6

Assume the values within each class are evenly spread.

A jar contains 5 red sweets, 4 blue sweets and 3 yellow sweets only. Priya takes three sweets at random from the jar without replacement.

A biased spinner lands on red with probability p, where p < 0.5. The spinner is spun twice. The probability that it lands on red exactly once is 0.42.

The ages, a years, of 120 people at a concert are summarised below.
0 < a ≤ 10: frequency 15
10 < a ≤ 25: frequency unknown; the frequency density of this class on a histogram is 2.4
25 < a ≤ 40: frequency 30
40 < a ≤ 70: frequency unknown